Geodesic
Stabilization of a~solution to the Cauchy problem for a~nondivergence parabolic equation with growing lower order coefficients
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 97-109.

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We obtain sharp sufficient conditions on the growth of lower order coefficients of a second-order parabolic equation under which a solution to the Cauchy problem stabilizes to zero uniformly in $x$ on every compact set $K\in\mathbb R^N$ in some classes of growing initial functions. 
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V. N. Denisov. Stabilization of a~solution to the Cauchy problem for a~nondivergence parabolic equation with growing lower order coefficients. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 97-109. http://geodesic.mathdoc.fr/item/TM_2010_270_a6/

[1] Krżyzański M., “Certaines inéqalités relatives aux solutions de l'equation parabolique linéaire normale”, Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. et Phys., 7:3 (1959), 131–135 | MR

[2] Krżyzański M., “Sur la solution fondamentale de l'équation linéaire normale du type parabolique dont be dernier coefficient est non borné. I, II”, Atti Accad. Naz. Lincei. Ser. 8. Rend. Cl. Sci. Fis. Mat. e Nat., 32 (1962), 326–330 | MR

[3] Eidelman S.D., Porper F.O., “O povedenii reshenii parabolicheskikh uravnenii vtorogo poryadka s dissipatsiei”, Dif. uravneniya, 7:9 (1971), 1684–1695 | MR

[4] Zhitomirskii Ya.I., “Klassy edinstvennosti i needinstvennosti resheniya zadachi Koshi dlya uravnenii 2-go poryadka”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, Vyp. 4, Izd-vo Kharkov. gos. un-ta, Kharkov, 1967, 87–115

[5] Smirnova G.N., “Zadachi Koshi dlya parabolicheskikh uravnenii, vyrozhdayuschikhsya na beskonechnosti”, Mat. sb., 70:4 (1966), 591–604 | MR | Zbl

[6] Murata M., “Non-uniqueness of the positive Couchy problem for parabolic equations”, J. Diff. Equat., 123:2 (1995), 343–387 | DOI | MR | Zbl

[7] Pinchover Y., “On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients”, Math. Ztschr., 223 (1996), 569–586 | DOI | MR | Zbl

[8] Denisov V.N., “O povedenii reshenii parabolicheskikh uravnenii pri bolshikh znacheniyakh vremeni”, UMN, 60:4 (2005), 145–212 | DOI | MR | Zbl

[9] Denisov V.N., “O stabilizatsii resheniya zadachi Koshi dlya parabolicheskogo uravneniya s mladshimi koeffitsientami”, Fund. i prikl. matematika, 12:4 (2006), 79–97 | MR

[10] Ilin A.M., Kalashnikov A.S., Oleinik O.A., “Lineinye uravneniya vtorogo poryadka parabolicheskogo tipa”, UMN, 17:3 (1962), 3–146 | MR | Zbl

[11] Ilin V.A., Sadovnichii V.A., Sendov B.Kh., Matematicheskii analiz, v. 1, Izd-vo MGU, M., 2004

[12] Fedoryuk M.V., Obyknovennye differentsialnye uravneniya, Nauka, M., 1985 | MR

[13] Vatson G.N., Teoriya besselevykh funktsii, Izd-vo inostr. lit., M., 1949

[14] Kondratev V.A., Landis E.M., “Kachestvennaya teoriya lineinykh differentsialnykh uravnenii v chastnykh proizvodnykh vtorogo poryadka”, Differentsialnye uravneniya s chastnymi proizvodnymi – 3, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 32, VINITI, M., 1988, 99–215 | MR

[15] Fridman A., Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968